Statistical Modeling for the Minimum Standby Supply Voltage of a Full SRAM Array

Transcription

1 Statistical Modeling for the Minimum Standby Supply Voltage of a Full SRAM Array Jiajing Wang 1, Amith Singhee, Rob A. Runtenbar, Benton H. Calhoun 1 1 University of Virginia, Charlottesville, VA Carnegie Mellon University, Pittsburg, PA CarnegieMellon 1

2 Outline Motivation SNM and DRV New DRV Model Based on SNM DRV Model Evaluation Conclusion

3 Motivation I: SRAM Leakage Power Savings & Stability Leakage power increases with scaling SRAM leakage power dominates Standby supply voltage (V DD ) scaling reduces leakage power effectively SRAM stability is degraded with V DD scaling Data Retention Voltage (DRV): min(v DD ) for preserving cell state 3

4 Motivation II: DRV Distribution of a Full SRAM Variations impact DRV of different cells V T variation has the strongest impact DRV is distributed on the same die DRV distribution has a heavier tail on right The tail is very important e.g. Histogram from a 5k-point Monte-Carlo (M-C) in 9nm Occurances DRV(mV) Tail sets min(v DD ) for the whole SRAM 4

5 Motivation III: Methods for DRV Tail Estimation An analytical model [Qin et al, ISQED4] for DRV of individual bitcells Complex and hard to find the tail value Full M-C simulation Accurate but too expensive for large SRAMs Small M-C simulation + extrapolation Inaccurate since DRV is a non-gaussian distribution Neither normal nor log-normal This work: Two fast and accurate methods A new DRV model based on SNM Statistical Blockade tool [Singhee et al, DATE7] 5

6 Outline Motivation SNM and DRV New DRV Model Based on SNM DRV Model Evaluation Conclusion 6

7 SNM & DRV I QB (mv) 1 DRV=65mV * Butterfly curves of the balanced/imbalanced cell at various V DD 65 V M (a) 65 1 Q (mv) QB (mv) 13 V M (b) 13 Q (mv) Static Noise Margin (SNM) the metric of stability SNMH: upper-left square SNML: lower-right square SNM = min(snmh, SNML) DRV=13mV DRV = V DD (when SNM=) 7

8 SNM & DRV II SNMH/SNML/SNM statistics SNMH/SNML is approximately normally distributed N(μ,σ ) SNMH & SNML are approximately identically distributed SNM statistics can be obtained from SNMH/SNML statistics [Calhoun et al, JSSC6] How the distribution of SNMH changes with V DD mean value μ moves, but the shape (i.e. σ) almost keeps same Occurrence V DD =.1V.V.3V.4V SNM High (mv) 8

9 SNM & DRV III SNM High (mv) 3 1 w/o 3σ 3σ 6σ 6σ V DD (mv) SNMH vs. V DD with/without mismatch in 1 FET almost linear before hitting zero (DRV point) approximately constant slope regardless of mismatch slope (k) extracted from DC sweep simulation 9

10 Outline Motivation SNM and DRV New DRV Model Based on SNM DRV Model Evaluation Conclusion 1

11 Derive DRV Model from SNM P N(μ,σ DD =V SNMH pick an initial V DD (V ); get the statistics (μ,σ ) of SNMH at V from M-C sim SNMH slope=k V DD get the linear dependency (k) of SNMH on V DD from DC sweep P N(μ,σ DD =x SNMH get SNMH statistics (normally distributed) at V DD point x: PDF: f SNMH and CDF: F SNMH with μ = μ +k(x-v ); σ = σ 11

12 Derive DRV Model from SNM Area1 P V DD =x SNM= SNM assume SNMH & SNML are i.i.d. SNM = min(snmh, SNML); so SNM CDF is (from order statistics): F SNM = F SNMH (F SNMH ) P Area1 corresponds to Area DRV = V DD (SNM=) P = DD ( DRV x) = 1 P( SNM, V x) Area DRV=x DRV get DRV statistics from SNM statistics 1

13 13 ) ( 4 1 ) ( 1 ) ( = σ μ σ μ V x k erfc V x k erfc x F DRV ( ) ( ) ) ( V x erfc k x F DRV + = μ σ Parameters k: the slope of SNMH vs. V DD V : the initial supply voltage for the small M-C sim μ, σ : the mean and standard deviation of SNMH when V DD =V erfc( ): the complementary error function SNM & DRV Model CDF of SNM distribution when V DD =x () (3) inverse CDF of DRV distribution CDF of DRV distribution ) ( 4 1 ) ( ), ( = = σ μ σ μ V x k s erfc V x k s erfc x V s SNM P DD (1)

14 Steps How to Use Our Model 1. extract k from a DC-sweep of SNM/SNMH vs. V DD. pick V, extract μ & σ from a 1.5~5K-point M-C simulation for SNMH 3. use eq.(1) to calculate P(SNM s) at some V DD point x or 4. use eq.() to calculate P(DRV x) or 5. use eq.(3) to calculate the V DD that is necessary to ensure that P(DRV V DD )=x 14

15 How to Use Our Model One example: Memory failure probability with V DD scaling Probablity(SNM ) k =.45 V = 1mV μ =11.mV σ = 9.3mV (17, 1 5 ) V DD (mv) when V DD =17mV, failure probability is 1-5 DRV for a 1Kb memory is 17mV If memory must tolerate some noise margin (e.g. mv) Use s=mv in eq.(1) Redefine DRV=V DD (SNM mv) 15

16 Outline Motivation SNM and DRV New DRV Model Based on SNM DRV Model Evaluation Conclusion 16

17 DRV Model Evaluation Compared with an alternate fast approach, the Statistical Blockade (SB) tool [Singhee et al, DATE7], up to 8σ Compared with Monte-Carlo up to 6σ 17

18 SB Tool & its application for DRV SB is a fast M-C simulation tool for rare events Perform initial sampling Build a classifier to filter samples prior to simulation Simulate only those points that are classified as tail points SB also builds a model for tail Fit true tail points to the Generalized Pareto Distribution (GPD) Estimate longer tails with GPD model Advantage Generic, i.e. can be used for any tail statistics Succeeded in previous tests on SRAM and flipflop This work: New application to DRV 18

19 DRV Model Comparison Worst DRV (mv) New model Blockade tool Normal Lognormal MonteCarlo Memory size σ Compared with M-C Average error rate 1.3% out to 6σ 1 5 speedup at 6σ Compared with SB Agreement up to 8σ 1 faster 19

20 DRV Model Evaluation II Fitting parameters of new model k: the slope of SNMH vs. V DD V : the initial supply voltage μ : the mean of SNMH when V DD =V σ : the standard deviation of SNMH when V DD =V N: the number of M-C sample points for extracting SNMH statistics To what extent does the model depend on parameter selection?

21 DRV Model Sensitivity I When N changes from 1k to 5k, the average error rate over Monte-Carlo is <3% 5 Average Error (%) Number of sample points for SNMH (N) 1

22 DRV Model Sensitivity II When k, µ, σ, or V varies, the average error rate over Monte-Carlo is <6% Average Error (%) 5 (a) k V =1 μ =11 σ =9.3 Average Error (%) 3 1 (b) μ (mv) V =1 σ =9.3 k=.45 Average Error (%) 3 (c) 1 V =1 μ =11 k= σ (mv) Average Error (%) 4 (d) V (mv) k=.45

23 Outline Motivation SNM and DRV New DRV Model Based on SNM DRV Model Evaluation Conclusion 3

24 Conclusion DRV of the SRAM cells is distributed Caused by within-die variations (i.e. mismatch) The tail point determines the minimum V DD Two new methods are proposed to model DRV tail A new model based on the connection of DRV and SNM The Statistical Blockade tool, applied to DRV for the first time Model accuracy and speed Compared with M-C (up to 6σ), avg. error rate is <% for model and SB tool New model is highly consistent with SB tool (up to 8σ) New model is insensitive to parameter fluctuations Compared with M-C for 1G-b memory, new model offers ~1 5 x speedup and SB tool offers ~1 4 x 4

25 Conclusion Model application: A canary-replica feedback scheme for standby V DD scaling in SRAM [Wang et al, CICC7] Estimate SRAM DRV tail Estimate P(DRV SRAM <DRV canary ) to configure canary cells Allows closed loop standby power management P Less power SRAM cell Failure Threshold More reliable Multiple sets of Canary cells 18Kb SRAM ARRAY Canary Replica & Test Circuit DRV 5

26 Thank You! Q & A 6

27 DRV Reduction Techniques Local mismatch impacts DRV most Use larger transistor sizes, which reduce the spread of the local threshold voltage variation Global P/N fet strength mismatch also impacts DRV Move a N/P strong process towards being balanced by using adaptive body biasing Bitline leakage impacts DRV significantly when mismatch occurs on the access transistor Use negative wordline or floating Bitline, etc., to reduce bitline leakage 7